The couple have not been seen together in public since the Invictus Games in Toronto, where they put on an affectionate display in front of media.
The actress is now free from her contractual obligations to Suits, the TV drama series, having filmed her last episode, and is said to be moving out of her Toronto home.
A move to London, where she is presumed to be staying with the Prince at Kensington Palace, has been interpreted as the clearest sign yet of their committed relationship.
"It seems as though Harry will pop the question any day now and we think a royal wedding is mostly likely next summer after the birth of William and Kate's third child," said Harry Aitkenhead, of bookmaker Coral.
Kensington Palace has given no sign that an announcement is imminent.
Surge in bets for a May wedding and will it be at Windsor, St Paul's or Westminster Abbey?
Bookmakers may have frozen bets on a royal engagement but there has been a plethora of wagers on when and where they may marry.
It is 1/2 on that they will walk down the aisle in 2018.
May appears to be the favourite month and money is split between Windsor Castle or Westminster Abbey as a venue.
It could also be held at St Paul's, where Harry's parents Charles and Diana married in 1981.
Other wedding-related wagers include family friend Elton John, who sang at Princess Diana's funeral, to sing at the wedding at 7/2.
With dubstep act Skream reported as a favoured band of Harry's, they are at 10/1 to perform at the wedding, with rap star Drake, who is from Meghan's home town of Toronto, priced at 20/1.
British fashion house Alexander McQueen is favourite to supply her attire for the big day and Meghan is 1/50 on to wear white.
Prince Harry is most likely to have his brother William alongside him as best man at 1/100 on while Prince George is at 500/1.
Paddy Power are also offering odds on Meghan being pregnant when they marry at 14/1 or already having a baby together at 33/1.